Complex dynamical networks: from measures to models Alain Barrat CPT, Marseille, France & ISI, Turin, Italy http://www.cpt.univ-mrs.fr/~barrat http://www.cxnets.org http://www.sociopatterns.org
Infrastructure networks Biological networks Communication networks Social networks Virtual networks ... • Empirical study and characterization: find generic characteristics (small-world, heterogeneities, hierarchies, communities...) , define DATA statistical characterization tools • Modeling: understand formation mechanisms • Consequences of the empirically found properties on dynamical phenomena taking place on the networks (epidemic spreading, robustness and resilience, etc…)
Dynamical networks Networks= (often) dynamical entities •Which dynamics? •Characterization? •Modeling? •Consequences on dynamical phenomena? (e.g. epidemics, information propagation…) Back to square one: Fundamental issue = data gathering!!!
Outline • Infrastructure networks –Empirics –Stationarity and dynamics • Human contact networks –Measuring infrastructure –Empirical data –A (simple) model –Dynamical processes
Example of dynamical infrastructure network: Cattle movements Bajardi et al, PLoS ONE (2011)
i w ij w ji j
Aggregate movements within a time window [ n ∆ t, ( n + 1) ∆ t ] ∆ t = 1 ... t t+1 t+2 t+3 time => Time ordered series of directed networks between farms
Stationary statistical properties
Stationary statistical properties Statistical stationarity P(k in/out ), P(s in/out ), P(w), ecc... of global distributions
Dynamic behavior of the network
Dynamic behavior of the network Need to take into account the full dynamical dataset, aggregated views can be misleading
Dynamic behavior of the network Lifetime distribution
Fluctuations Fluctuations of daily/weekly nodes’ strengths Bajardi et al, PLoS ONE (2011)
Consequences of temporal fluctuations Ex: percolation analysis • used as probe of networks • identification of most important nodes • definition of strategies for disease containment network at time T 1 network at time T 2
Percolation analysis network at time T 1 network at time T 2 targeted nodes removal targeted T 1 nodes removal
Percolation analysis network at time T 1 network at time T 2 targeted nodes removal targeted T 1 nodes removal
Percolation analysis network at time T 1 network at time T 2 targeted nodes removal targeted T 1 nodes removal
Percolation analysis network at time T 1 network at time T 2 targeted nodes removal targeted T 1 nodes removal
Percolation analysis network at time T 1 network at time T 2 Targeted attack based on static/ targeted nodes removal targeted T 1 nodes removal past measures is ineffective Ex: monthly networks n=3 and n=4 fraction of nodes removed (order= decreasing degree in 3rd monthly network)
New approaches combining dynamics • on the network • of the network • to study dynamical processes on dynamical networks • to define importance/centrality of nodes • to define surveillance strategies Bajardi et al., J. Roy. Soc. Interface (2012)
Dynamical networks of human interactions
Data on the dynamics of human interaction networks • Mobile phones (Onnela et al 2007, Gonzalez et al 2009,...) – Localisation, mobility patterns, predictability – Strength of weak ties – ... • Social interaction networks – Bluetooth, wifi (O’ Neill et al 2006; Scherrer et al 2008; Eagle, Pentland 2009) – MIT Reality mining project (sociometric badges) – MOSAR european project (hospitals) – Salathé et al. 2010 (highschool) – ...
Gathering data: The SocioPatterns collaboration what are the statistical and dynamical properties of the networks of contact and co-presence of people in social interaction? fine-grained spatial (~ m) and temporal (<min) resolution
Motivations ★ fundamental knowledge on human contact ★ epidemiology ★ social sciences ★ ad-hoc networks ★ integration with on-line information ★ ...
• 2.4 GHz microwave ISM band • PIC microcontroller + Nordic RF chip • 128 bytes RAM, 2 kB flash program memory • 1 coin cell battery • 15-20$ / unit RFID tag OPEN HARDWARE AND FIRMWARE
Contact detection 10 Short distance (~1-2m): Exchange of very low power data packets 42 “42 saw 10 at power 0” •Two power levels => 2 detection ranges • Face to face situation • Statistical detection => 20s time resolution • Small, • Scalable
dynamical network of f2f proximity http://www.vimeo.com/6590604
DATE EVENT SIZE DURATION May 2008 Socio-physics workshop, Torino, IT ~65 3 days Jun 2008 ISI offices, Torino, IT ~25 3 weeks Oct 2008 ISI workshop, Torino, IT ~75 3 days Dec 2008 Chaos Comm. Congress, Berlin, DE ~600 4 days Apr-Jul 2009 Science Gallery, Dublin, IE ~30,000 3 months Jun 2009 ESWC09, Crete, GR ~180 4 days Jun 2009 SFHH, Nice, FR ~360 2 days Jul 2009 ACM HT2009, Torino, IT ~120 3 days Oct 2009 Primary school, Lyon, FR ~250 2 days Nov 2009 Bambino Gesù Hospital, Rome, IT ~250 10 days Jun 2010 ESWC10, Crete, GR ~200 4 days Apr 2010 Practice Mapping, Gijon, ES ~100 10 days Jul 2010 H-Farm, Treviso, IT ~200 6 weeks
>a glimpse of data Several data sets available at www.sociopatterns.org
>school
contacts in a primary school J. Stehle, et al. High-Resolution Measurements of Face-to-Face Contact Patterns in a Primary School PLoS ONE 6(8), e23176 (2011)
School cumulative f2f network (2 days, 2 min threshold) J. Stehlé et al. PLoS ONE 6(8):e23176 (2011) •Epidemiology: •Information of models •Design and efficiency of containment measures •Social sciences: •Gender segregation •Age homophily
class contact matrix J. Stehle, et al. High-Resolution Measurements of Face-to-Face Contact Patterns in a Primary School PLoS ONE 6(8), e23176 (2011)
>hospital
children doctors C N D A P nurses parents auxiliaries
class-level contact networks A A − ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● A − D D D − ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● A − N D − N N N − ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● A − P D − P N − P P P − ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● A − C D − C C − N C − P C C −
Epidemiology: Contact matrices
>dealing with data: similarities and differences across contexts
DATE EVENT SIZE DURATION May 2008 Socio-physics workshop, Torino, IT ~65 3 days Jun 2008 ISI offices, Torino, IT ~25 3 weeks Oct 2008 ISI workshop, Torino, IT ~75 3 days Dec 2008 Chaos Comm. Congress, Berlin, DE ~600 4 days Apr-Jul 2009 Science Gallery, Dublin, IE ~30,000 3 months Jun 2009 ESWC09, Crete, GR ~180 4 days Jun 2009 SFHH, Nice, FR ~400 2 days Jul 2009 ACM HT2009, Torino, IT ~120 3 days Oct 2009 Primary school, Lyon, FR ~250 2 days Nov 2009 Bambino Gesù Hospital, Rome, IT ~250 10 days Jun 2010 ESWC10, Crete, GR ~200 4 days Apr 2010 Practice Mapping, Gijon, ES ~100 10 days Jul 2010 H-Farm, Treviso, IT ~200 6 weeks
Different contexts • Conference (HT09) – Fixed number of attendees – Unconstrained mobility • Museum (SG) – Flux of individuals – Predefined visiting path • School Similarities/differences in the f2f proximity patterns? 41 L. Isella et al. , Journal of Theoretical Biology 271, 166 (2011)
Daily cumulated networks Conference Museum Non small-world School Small-world Communities
cumulative contact networks • color encodes the time of day • node are colored by arrival time • several groups (guided tours)
Exp. degree distributions HT09 SG Museum Conference
Similar contact durations distributions 0 10 SG -1 HT09 10 SFHH ESWC09 -2 10 ESWC10 Hospital P( ∆ t ij ) -3 Highschool 10 -4 10 -5 10 -6 10 1 2 3 4 10 10 10 10 ∆ t ij Contact duration
Weight (cumulative contact time) distributions
Different “superspreading” patterns Conference Museum Superspreading Opposite trend k=number of distinct persons contacted s=total time spent in contact Random weights: s ~ <w>k L. Isella et al. , Journal of Theoretical Biology 271, 166 (2011)
>how to go beyond?
“synopsis” of dynamic network data A 38.5 0.2 7.8 0.4 1.0 D 0.2 3.8 1.0 0.5 0.2 N 3.1 1.2 12.9 0.9 1.0 P 0.1 0.2 0.4 0.0 11.3 C 0.2 0.3 0.5 15.3 0.3 A D N P C x (x,y) ?
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